#4111280

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,15)(11,12,16,14)(18,19)(20,24)(25,26,30,28)(27,29,32,31);(2,3)(6,7)(9,11)(10,14)(12,15)(13,16)(20,24)(22,23)(27,32)(29,31);(4,8)(6,7)(10,15)(11,16)(17,18,21,19)(20,22,24,23)(25,27,30,32)(26,31,28,29);(1,3,5,2)(4,6,8,7)(9,12,13,14)(10,16,15,11)(17,19,21,18)(20,23,24,22)(25,28,30,26)(27,29,32,31);(1,4)(2,6)(3,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4018476);